It may depend on what time the ship sails. I had e-mailed Celebrity about this for our 6pm sailing, they also said 2pm. I then sent another e-mail explaining I would like to drop off the luggage earlier so I can explore the city before boarding. They said I can drop off luggage as early as 11 am. So I'm aiming to be at the port by then and hopefully they'll let us board that early.
If you post what time your ship sails and out of what port, maybe you'll get responses from those that have done the same thing.